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The Change-of-Coordinate Matrix From the Polytabloid Basis to the Web Basis -
The symmetric group, I.e. the set of permutations on a finite set equipped with a group structure, admits a matrix representation where each permutation gets assigned a matrix. Each of such matrices can be interpreted as a linear map on a fixed vector space, although the choice of basis is not unique. In fact, two representations are said to be isomorphic if the corresponding matrices are conjugates via the change-of-coordinate matrix. The key question here is to study properties of the unique base-change matrix for two famous bases. In joint work with M. S. Im, we proved that the matrix has positive entries in the upper-triangular portion. The properties of the inverse matrix remain largely open.
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